Yes, Einstein's theory of gravity (called General Relativity), which is the theory which predicts black holes, also describes the interior of black holes equally well.
Probably a good analog to thinking about the behaviour of light near a black hole is to think of sound waves in a fluid (say water) which at some point flows faster than the speed of sound. That surface where the velocity of the fluid exceeds the velocity of sounds, behaves just like the horizon of a black hole in many respects. The equations of motion of the sound wave are very similar to those of light near a black hole.
In the sound case, you know that if you were a fish who was unfortunate enough to be dragged through that surface, then the behaviour of sound in your vicinity would not seem to be unusual at all. Sound would still travel away from you at the speed of sound. It is just an observer at infinity who would notice that no sound that you emitted after you fell through that surface would ever reach him.
In the case of the sound in the fluid, you can ascribe this strange behaviour to the flow of the fluid. In the case of a black hole, the strange behaviour of light is due to the change in geometry of the spacetime, but the two effects are the same. (In fact the sound waves behave exactly as though they too were in a changed geometry, caused in that case by the fluid flow).
Just as the equations of fluid flow (hydrodynamics) apply equally well both inside and outside that horizon, so do Einstein's equations apply equally well both inside and outside the horizon. Just as for the fish falling in, nothing strange happens at or beyond the sonic horizon, so for an observer falling into the black hole nothing strange happens at or beyond the black hole horizon.
This analogy is not to say that there is really an aether, whose flow makes black holes. That part (the cause of the change in geometry) is not a good part of the analogy. In the light and black hole case, the equations of General relativity are equations which govern the behaviour of the metric directly, rather than via the intermediary of the fluid in the sonic case.
[Editor: and in any case, the speed of light remains constant.]
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